We present the results of the measured shapes of 832 galaxies in 11 galaxy clusters at 1.0 < z < 1.4 from the GOGREEN survey. We measure the axis ratio (q), the ratio of the minor to the major axis, of the cluster galaxies from near-infrared Hubble Space Telescope imaging using Sersic profile fitting and compare them with a field sample. We find that the median q of both star-forming and quiescent galaxies in clusters increases with stellar mass, similar to the field. Comparing the axis ratio distributions between clusters and the field in four mass bins, the distributions for star-forming galaxies in clusters are consistent with those in the field. Conversely, the distributions for quiescent galaxies in the two environments are distinct, most remarkably in 10.1 <= log(M/M-circle dot)<10.5 <i where clusters show a flatter distribution, with an excess at low q. Modelling the distribution with oblate and triaxial components, we find that the cluster and field sample difference is consistent with an excess of flattened oblate quiescent galaxies in clusters. The oblate population contribution drops at high masses, resulting in a narrower q distribution in the massive population than at lower masses. Using a simple accretion model, we show that the observed q distributions and quenched fractions are consistent with a scenario where no morphological transformation occurs for the environmentally quenched population in the two intermediate-mass bins. Our results suggest that environmental quenching mechanism(s) likely produce a population that has a different morphological mix than those resulting from the dominant quenching mechanism in the field.